Abstract
Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean d-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.
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Notes
We use uppercase letters to indicate specially choosen \((d-2)\)-dimensional faces, i.e., faces which will appear again in subsequent constructions; and we use lowercase letters to indicate “arbitrary” \((d-2)\)-dimensional faces.
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Acknowledgements
The author is grateful to Dr. Evgeniĭ P. Volokitin for assistance in preparation of the figures and to Prof. Alexey Yu. Kokotov for bringing his attention to [35].
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Alexandrov, V. The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in \(\mathbb R^d\) does not always remain unaltered during the flex. J. Geom. 111, 32 (2020). https://doi.org/10.1007/s00022-020-00541-8
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DOI: https://doi.org/10.1007/s00022-020-00541-8
Keywords
- Flexible polyhedron
- Dihedral angle
- Volume
- Laplace operator
- Dirichlet eigenvalue
- Neumann eigenvalue
- Weyl’s law
- Weyl asymptotic formula for the Laplacian
- Asymptotic behavior of eigenvalues